Про будову одного класу груп з умовою щільності нормальності для неперіодичних неабелевих підгруп

Abstract

Groups with large systems of normal subgroups are a rather old object of research in group theory. The presence of a large number of normal subgroups greatly affects the structure of the group. For example, if all subgroups of the group are normal, then non-Abelian groups with this property have a very simple structure, as shown by the results of works [1, 2]. From these works of R. Dedekind and R. Behr, the study of arbitrary groups G began, in which some system of subgroups Σ of the group G satisfies the condition of normality. This direction is one of the important ones in group theory. Its main purpose is to describe generalizations of dedekind groups. One of such generalizations is carried out by narrowing the system of subgroups Σ that are normal in the whole group. The named generalization of dedekind groups can be found in the works of many authors. In 1968, A. Mann [3] began to study groups in which not all subgroups of the system Σ are normal, but those groups G that have a normal subgroup N placed between any two subgroups A and B of Σ, where A is a proper is a non-maximal subgroup of B. It has Σ – the system of all subgroups of group G. The groups introduced by A. Mann and S. M. Chernikov in 1975 called them groups with the density condition of normality for all subgroups. He also introduced the concept of density conditions for any group-theoretic property V (complementarity, subnormality, near-normality, etc.) of the system of subgroups Σ [4, Chapter 7]. We will say that a group G has a dense system of normal Σ-subgroups if, for any pair of subgroups A < B such that A is not maximal in B, there exists a normal subgroup N in G located between A and B, that is, A ≤ N ≤ B. If Σ is the system of all subgroups of the group G, then we obtain the definition of groups with normality density conditions for all subgroups (USCHN[ ]-groups). Locally stepped of this kind are described in works [5 – 14]. We will say that a non-Abelian group G has a dense system of normal non- periodic non-Abelian subgroups if for any such pair of non-periodic non-Abelian subgroups A<B such that A is not maximal in B, there exists a normal subgroup N in G located between A and B, i.e. A ≤ N ≤ B (USCHN[NN]-group).